|Project heads||Martin Eigel, Dietmar Hömberg, Rene Henrion and Reinhold Schneider|
|Staff||Johannes Neumann, Thomas Petzold|
D-OT1 (Hintermüller, Mielke, Surowiec, Thomas)
C-SE4 (Kraus, Kutyniok, Wagner)
C-SE7 (Schoenmakers, Spokoiny)
D-SE9 (Schneider, Tröltzsch)
Ch. Ruprecht, L. Blank (Universität Regensburg)
H. Matthies, E. Zander (TU Braunschweig)
|Industrial Cooperation||TEMBRA GmbH Wind Turbine Development|
Introduction The application focus of this project is the topology optimization of the main frame of wind turbines which is the central assembly platform at the tower head. It is a crucial component which has to be designed as light as possible but, at the same time, as rigid as necessary. In order to fully exploit the potential optimizations regarding the design, uncertainties regarding the impurities of the material (cast iron) and the forces acting on the structure have to be taken into account.
A main novelty of this project is that it combines a phase field relaxed topology optimization problem not only with uncertain loading and material data but also with chance state constraints. The latter are introduced to avoid non-admissible large stresses in the structure, a critical problem when engineering such structures.
The work in this project considers recent results on stochastic topology optimization with uncertain data. However, our approach differs in either the stochastic discretization approach or the optimization formulation or both and introduces an alternative phase field based optimization and additional constraints based on finite dimensional stochastic optimization.
Topology Optimization The goal in topology optimization is to partition a given domain into regions occupied by either void or material by minimizing a given cost functional \(J_1(\boldsymbol u,\varphi)\) such that the displacement \(\boldsymbol u\) solves a mechanical equilibrium problem. The material distribution is described with the help of a phase field variable \(\varphi\), taking values in \([0,1]\). To avoid homogenized microstructures, a perimeter penalization \(J_2(\varphi)\) is added to the cost functional.
Uncertainty Quantification The incorporation of uncertain parameters in PDE simulations has evolved as one of the most active emerging fields in applied mathematics in recent years. Apart from classical Monte Carlo techniques, methods based on the expansion of stochastic fields in appropriate bases related to a probability space have received great interest in the community. This can be attributed to potentially high convergence rates, possible adaptivity of the expansion and versatility regarding the application areas.
The most popular techniques are stochastic collocation (SC) and stochastic Galerkin FEM (SGFEM). Reliable averaging has only been shown for SGFEM as yet but is also envisaged as part of this project in the context of SC methods. The inclusion of uncertainties comes at the expense of much larger algebraic problems for numerical simulations. Possible remedies for the exponential growth of complexity are model reduction techniques such as hierarchical tensor representations. Such approximations will also be investigated in this project, see [Eigel, Pfeffer & Schneider, 2015] for first results.